The conversation continued, and I was about to capitalize the D in the second instance of “Dad”; but I realized I had to look up this basic capitalization issue. Cambridge Advanced Learners tells me this:

dad: noun
a father

It was lovely to see your mum and dad at the school concert last night.

[as form of address] Can you give me a lift back from the cinema tonight, Dad?

The American English definition of the word is the same, except for the examples: “Can you give me $20, Dad?” and “My dad’s a fireman.” (Hmm. The British version talks about what Dad is doing or has done, the American one about what Dad’s job is—and oh, yeah, can you give me money, Dad? Really, Learners? Can we be any more stereotypical?)

In other words, parental titles are treated like Mister, Missus, Ms. (which has no long form—an abbreviation without a word it’s abbreviating) and also earned titles like Doctor, when and only when they’re being used as titles. When “Dad” or “Mum” is being used as a descriptor (think “my dear father is a horrible man who beats me”) it’s not appropriate to capitalize the word. Think of it like a title: When addressing a mother or father, use the words like titles. When referring to them in a specific or general way, treat the words as standard nouns,